Homomorphic Mappings for Value-Preserving State Aggregation in Markov Decision Processes

State aggregation aims to reduce the computational complexity of solving Markov Decision Processes (MDPs) while preserving the performance of the original system. A fundamental challenge lies in optimizing policies within the aggregated, or abstract, space such that the performance remains optimal in the ground MDP-a property referred to as {"}optimal policy equivalence {"}. This paper presents an abstraction framework based on the notion of homomorphism, in which two Markov chains are deemed homomorphic if their value functions exhibit a linear relationship. Within this theoretical framework, we establish a sufficient condition for the equivalence of optimal policy. We further examine scenarios where the sufficient condition is not met and derive an upper bound on the approximation error and a performance lower bound for the objective function under the ground MDP. We propose Homomorphic Policy Gradient (HPG), which guarantees optimal policy equivalence under sufficient conditions, and its extension, Error-Bounded HPG (EBHPG), which balances computational efficiency and the performance loss induced by aggregation. In the experiments, we validated the theoretical results and conducted comparative evaluations against seven algorithms.